ABSTRACT
The authors will classify all finite fields and also explore splitting fields of polynomials. They address in this investigation is whether there are finite fields other than for p prime. They can actually explicitly determine all the finite fields. Now that they have proved Theorem, they can determine the groups U n of the units in. They will return to the question of how many fields there are of a given order p n. It may not be surprising that any two splitting fields of a polynomial over a field F are isomorphic, and they will finish this investigation by verifying this fact. Since any fields of order p n for a prime p and a positive integer n are splitting fields for, this will show that any two finite fields of the same order are isomorphic.
